On two conditional forms of the equation \(\| f(x+y)\|=\| f(x)+f(y)\|\)
type: Item
instance of: scholarly article
title: On two conditional forms of the equation \(\| f(x+y)\|=\| f(x)+f(y)\|\)
zbMATH Open document ID: 0776.39003
DOI: 10.1007/BF01855876
author: Fulvia Skof
published in: Aequationes Mathematicae
publication date: 1993-11-22T00:00:00Z
full work available at URL: https://eudml.org/doc/137519
review text: Let \(X\) and \(E\) be real linear spaces and let \(E=(E,\|\circ\|)\) be a strictly normed space, that is, if \(u,v\in E\), \(u\neq 0\), \(v\neq 0\) and \(\| u+v\|=\| u\|+\| v\|\), then there exists a real \(t>0\) such that \(u=tv\). Let \(f\) be a function from \(X\) to \(E\). Without any regularity conditions on \(F\), the author shows that (a) \((x,y)\in X\times X\), \(f(x+y)\neq 0\Rightarrow\| f(x+y)\|=\| f(x)+f(y)\|\) (b) \((x,y)\in X\times X\), \(f(x)+f(y)\neq 0\Rightarrow\| f(x+y)\|=\| f(x)+f(y)\|\) (c) \(f(x+y)=f(x)+f(y)\) for all \((x,y)\in X\times X\) are equivalent.
Mathematics Subject Classification ID: 39B52
zbMATH DE Number: 202380
zbMATH Keywords: conditional forms
zbMATH Keywords: real linear spaces
zbMATH Keywords: strictly normed space
reviewed by: Harkrishan L. Vasudeva
MaRDI profile type: Publication
cites work: Isometries in Normed Spaces
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description: scientific article; zbMATH DE number 202380